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Theorem nb3grprlem1 28328
Description: Lemma 1 for nb3grpr 28330. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nb3grpr.v 𝑉 = (Vtx‘𝐺)
nb3grpr.e 𝐸 = (Edg‘𝐺)
nb3grpr.g (𝜑𝐺 ∈ USGraph)
nb3grpr.t (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
Assertion
Ref Expression
nb3grprlem1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nb3grpr.s . . . . . . 7 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
2 prid1g 4721 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐵, 𝐶})
323ad2ant2 1134 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵 ∈ {𝐵, 𝐶})
41, 3syl 17 . . . . . 6 (𝜑𝐵 ∈ {𝐵, 𝐶})
54adantr 481 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6 eleq2 2826 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
76eqcoms 2744 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
87adantl 482 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
95, 8mpbid 231 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10 nb3grpr.g . . . . . 6 (𝜑𝐺 ∈ USGraph)
11 nb3grpr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211nbusgreledg 28301 . . . . . . 7 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13 prcom 4693 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
1413a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
1514eleq1d 2822 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
1612, 15bitrd 278 . . . . . 6 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1710, 16syl 17 . . . . 5 (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1817adantr 481 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
199, 18mpbid 231 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20 prid2g 4722 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐵, 𝐶})
21203ad2ant3 1135 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐵, 𝐶})
221, 21syl 17 . . . . . 6 (𝜑𝐶 ∈ {𝐵, 𝐶})
2322adantr 481 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24 eleq2 2826 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2524eqcoms 2744 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2625adantl 482 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2723, 26mpbid 231 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
2811nbusgreledg 28301 . . . . . . 7 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29 prcom 4693 . . . . . . . . 9 {𝐶, 𝐴} = {𝐴, 𝐶}
3029a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
3130eleq1d 2822 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
3228, 31bitrd 278 . . . . . 6 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3310, 32syl 17 . . . . 5 (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3433adantr 481 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3527, 34mpbid 231 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
3619, 35jca 512 . 2 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37 nb3grpr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
3837, 11nbusgr 28297 . . . . 5 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
3910, 38syl 17 . . . 4 (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
4039adantr 481 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41 nb3grpr.t . . . . . . . . . 10 (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
42 eleq2 2826 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4341, 42syl 17 . . . . . . . . 9 (𝜑 → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4443adantr 481 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45 vex 3449 . . . . . . . . . . 11 𝑣 ∈ V
4645eltp 4649 . . . . . . . . . 10 (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶))
4711usgredgne 28154 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴𝑣)
48 df-ne 2944 . . . . . . . . . . . . . . . . 17 (𝐴𝑣 ↔ ¬ 𝐴 = 𝑣)
49 pm2.24 124 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5049eqcoms 2744 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5150com12 32 . . . . . . . . . . . . . . . . 17 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5248, 51sylbi 216 . . . . . . . . . . . . . . . 16 (𝐴𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5347, 52syl 17 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5453ex 413 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5510, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5655adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5756com3r 87 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
58 orc 865 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣 = 𝐵𝑣 = 𝐶))
59582a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
60 olc 866 . . . . . . . . . . . 12 (𝑣 = 𝐶 → (𝑣 = 𝐵𝑣 = 𝐶))
61602a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6257, 59, 613jaoi 1427 . . . . . . . . . 10 ((𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6346, 62sylbi 216 . . . . . . . . 9 (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6463com12 32 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6544, 64sylbid 239 . . . . . . 7 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6665impd 411 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵𝑣 = 𝐶)))
67 eqid 2736 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
68673mix2i 1334 . . . . . . . . . . . . . . . . 17 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
691simp2d 1143 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑌)
70 eltpg 4646 . . . . . . . . . . . . . . . . . 18 (𝐵𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7169, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7268, 71mpbiri 257 . . . . . . . . . . . . . . . 16 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
7372adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7574bicomd 222 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7675adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7773, 76mpbid 231 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
7842bicomd 222 . . . . . . . . . . . . . . . 16 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7941, 78syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8079adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8177, 80mpbid 231 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐵) → 𝑣𝑉)
8281ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐵𝑣𝑉))
8382adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵𝑣𝑉))
8483impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
85 preq2 4695 . . . . . . . . . . . . . . 15 (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
8685eleq1d 2822 . . . . . . . . . . . . . 14 (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8786eqcoms 2744 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8887biimpcd 248 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
8988ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
9089impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
9184, 90jca 512 . . . . . . . . 9 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
9291ex 413 . . . . . . . 8 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93 tpid3g 4733 . . . . . . . . . . . . . . . . . 18 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
94933ad2ant3 1135 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
951, 94syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
9695adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9897bicomd 222 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
9998adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
10096, 99mpbid 231 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
10179adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
102100, 101mpbid 231 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐶) → 𝑣𝑉)
103102ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐶𝑣𝑉))
104103adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶𝑣𝑉))
105104impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
106 preq2 4695 . . . . . . . . . . . . . . 15 (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107106eleq1d 2822 . . . . . . . . . . . . . 14 (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108107eqcoms 2744 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109108biimpcd 248 . . . . . . . . . . . 12 ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110109ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111110impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112105, 111jca 512 . . . . . . . . 9 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113112ex 413 . . . . . . . 8 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11492, 113jaoi 855 . . . . . . 7 ((𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115114com12 32 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵𝑣 = 𝐶) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11666, 115impbid 211 . . . . 5 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵𝑣 = 𝐶)))
117116abbidv 2805 . . . 4 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)})
118 df-rab 3408 . . . 4 {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119 dfpr2 4605 . . . 4 {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)}
120117, 118, 1193eqtr4g 2801 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
12140, 120eqtrd 2776 . 2 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
12236, 121impbida 799 1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wne 2943  {crab 3407  {cpr 4588  {ctp 4590  cfv 6496  (class class class)co 7357  Vtxcvtx 27947  Edgcedg 27998  USGraphcusgr 28100   NeighbVtx cnbgr 28280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-upgr 28033  df-umgr 28034  df-usgr 28102  df-nbgr 28281
This theorem is referenced by:  nb3grpr  28330  nb3grpr2  28331  nb3gr2nb  28332
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